Circle patterns with the combinatorics of the square grid
Oded Schramm · Duke Mathematical Journal · 1997
. Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is infinite dimensional. In Particular, Doyle's conjecture is false in this setting. Mobius invariants of circle patterns are introduced, and turn out to be discrete analogs of the Schwarzian derivative. The invariants satisfy a nonlinear discrete version of the Cauchy-Riemann equations. A global analysis of the solutions of these equations yields a rigidity theorem characterizing the Doyle spirals. It is also shown that by prescribing boundary values for the Mobius invariants, and solving the appropriate Dirichlet problem, a locally univalent meromorphic function can be approximated by circle patterns. 1991 Mathematics Subject Classification. 30C99, 05B40, 30D30, 31A05, 31C20, 30G25. Key words and phrases. Meromorphic functions, Schwarzian derivative, rigidity, error function, Dirichlet ...